This question is already answered here, for both sin and Γ: mathoverflow.net/questions/19946/…. The answer is $O(M(n)\log n)$, where $M(n)$ is the complexity of multiplication.

One practical sufficient criterion is described in mathoverflow.net/questions/287091/…: the relevant I-indexed diagram should be fibrant in a model structure on D-indexed diagrams whose weak equivalences are created by the homotopy colimit functor.

@PraphullaKoushik: The Duistermaat–Kolk construction is explained in Section 1.14 of their book Lie Groups. The DG-manifolds point of view explains where formulas like $dω+[ω,ω]/2$ come from, and in the Fiorenza–Schreiber–Stasheff paper “Čech cocycle for differential characteristic classes” it is deployed in full generality to define connections for bundles over Lie ∞-groups.

There are several questions asked here, and they are somewhat vague. For instance, what is G? A Lie group, a topological group, or, perhaps, a Lie ∞-group? One possible definition of a local system on X is that it is a map X→LConst(S), where S is an ∞-groupoid, e.g., S=BG for an ∞-group G. In this case, G can be a group, more generally, an ∞-group. A Lie group G will not give a local system unless X→BG factors through LConst((BG)(pt))→BG, i.e., a local system with coefficients in the underlying discrete group of G.

The necessary and sufficient criterion for categorical equivalences is similar: the map $f$ should be weakly equivalent to a cartesian (or cocartesian) fibration $g$ (i.e., $f$ is an ∞-categorical Street fibration), and the fibers of $g$ should be contractible. Of course, if $f$ is already a (co)cartesian fibration, we can simply take the fibers of $f$.

Have you considered using Quillen's Theorem A, in Joyal's version for quasicategories? (For example, see Theorem 4.1.3.1 in Lurie's Higher Topos Theory.)

@Stabilo: Using a concrete description of $\def\Exi{\mathop{\sf Ex^∞}}\def\cC{{\cal C}}\Exi \cC$, you can interpret an element in the $n$th homotopy group of $\Exi \cC$ as a diagram in $\cC$ whose indexing category is given by the subdivided $n$-sphere (which is always a poset if you subdivide two or more times).

@Stabilo: Of course: $\def\Exi{\mathop{\sf Ex^∞}}\def\cC{{\cal C}}\Exi \cC$ can be shown to present the ∞-groupoid $\cC[\cC^{-1}]$, i.e., $\cC$ with all of its morphisms inverted up to homotopy. Even if $\cC$ is an ordinary category, $\cC[\cC^{-1}]$ is typically an ∞-groupoid, not a 1-groupoid. The $n$th homotopy group of this ∞-groupoid is precisely the group of isomorphism classes of n-morphisms in $\cC[\cC^{-1}]$ with source and target being identities on the basepoint object and the group structure being given by composition in $\cC[\cC^{-1}]$.

@LorenzoRiva: I was answering your question precisely as it was formulated in the last sentence (“That is, how closely…”), using $[-,-]$ as one of the standard notations for the enriched hom, i.e., taking limits of the functors you previously described. Of course, if you are interested in individual values of these functors instead, you can repeat the same Yoneda trick, deriving the enriched case from the nonenriched statement.

@FernandoMuro: The link should probably be posted as an answer, to ensure the question does not keep showing up on the home page.

@JasonStarr: It is the same notion, but I have not seen any claims (in the answer itself or anywhere in the published literature) that “étale space” is meant to be a literal translation of “espace étalé”.

@JasonStarr: The adjective “étale” in “étale space” is the same as “étale” in “étale map”, which is quite natural since the map from the étale space to the base space is indeed an étale map. There is no reason why “étale space” must be a literal translation of “espace étalé”. Also, the spelling “étale space” is far more common in the published literature (in English) than anything containing “étalé”; there any many books using “étale space”, but I am aware of only one book (Wells) using “étalé space”.